Sequences from chemical structures as Hadamard self-similar matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg68646] Sequences from chemical structures as Hadamard self-similar matrices*From*: Roger Bagula <rlbagula at sbcglobal.net>*Date*: Mon, 14 Aug 2006 06:44:25 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

My work on group theory representations taught me how to repartition arrays in Mathematica so that I could do Hadamard self-similarity in fairly large matrices. Although they aren't techically a new kind of fractal, no one before has been able to express chemical sturctures as matrices like this as far as I know. I thought of how to do it last night and got it working for 2by2's and 3by3's this morning. This 6by6 gives a crystal like matrix structure for Trigonal prisms. ( triangles like that in a spectral prism) These matrices appear to be both fractal and tile-like! Clear[M, v, a, aaa, t] t[n_, m_] := {{0, 1, 1, 1, 0, 0}, {1, 0, 1, 0, 1, 0}, {1, 1, 0, 0, 0, 1}, {1, 0, 0, 0, 1, 1}, {0, 1, 0, 1, 0, 1}, {0, 0, 1, 1, 1, 0}}[[n, m]] a = Table[t[n, m]*t[i, j], {n, 1, 6}, {m, 1, 6}, {i, 1, 6}, {j, 1, 6}] M = Flatten[Table[{Flatten[Table[a[[ n, m]][[1, i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[2, i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[ n, m]][[3, i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[4, i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[ n, m]][[5, i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[6, i]], {n, 1, 6}, {i, 1, 6}]]}, {m, 1, 6}], 1] ListDensityPlot[M, Mesh -> False] v[1] = Table[Fibonacci[n], {n, 0, 35}] v[n_] := v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 1, 50}] Det[M - x*IdentityMatrix[36]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[36]] == 0, x][[n]], {n, 1, 36}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]